<!DOCTYPE html><html><head>
      <title>09</title>
      <meta charset="utf-8">
      <meta name="viewport" content="width=device-width, initial-scale=1.0">
      
      <style>
      /*!
 * reveal.js
 * http://revealjs.com
 * MIT licensed
 *
 * Copyright (C) 2018 Hakim El Hattab, http://hakim.se
 */
/*********************************************
 * RESET STYLES
 *********************************************/
html, body, .reveal div, .reveal span, .reveal applet, .reveal object, .reveal iframe,
.reveal h1, .reveal h2, .reveal h3, .reveal h4, .reveal h5, .reveal h6, .reveal p, .reveal blockquote, .reveal pre,
.reveal a, .reveal abbr, .reveal acronym, .reveal address, .reveal big, .reveal cite, .reveal code,
.reveal del, .reveal dfn, .reveal em, .reveal img, .reveal ins, .reveal kbd, .reveal q, .reveal s, .reveal samp,
.reveal small, .reveal strike, .reveal strong, .reveal sub, .reveal sup, .reveal tt, .reveal var,
.reveal b, .reveal u, .reveal center,
.reveal dl, .reveal dt, .reveal dd, .reveal ol, .reveal ul, .reveal li,
.reveal fieldset, .reveal form, .reveal label, .reveal legend,
.reveal table, .reveal caption, .reveal tbody, .reveal tfoot, .reveal thead, .reveal tr, .reveal th, .reveal td,
.reveal article, .reveal aside, .reveal canvas, .reveal details, .reveal embed,
.reveal figure, .reveal figcaption, .reveal footer, .reveal header, .reveal hgroup,
.reveal menu, .reveal nav, .reveal output, .reveal ruby, .reveal section, .reveal summary,
.reveal time, .reveal mark, .reveal audio, .reveal video {
  margin: 0;
  padding: 0;
  border: 0;
  font-size: 100%;
  /* font: inherit; */
  vertical-align: baseline; }

.reveal article, .reveal aside, .reveal details, .reveal figcaption, .reveal figure,
.reveal footer, .reveal header, .reveal hgroup, .reveal menu, .reveal nav, .reveal section {
  display: block; }

/*********************************************
 * GLOBAL STYLES
 *********************************************/
html,
body {
  width: 100%;
  height: 100%;
  overflow: hidden; }

body {
  position: relative;
  line-height: 1;
  background-color: #fff;
  color: #000; }

/*********************************************
 * VIEW FRAGMENTS
 *********************************************/
.reveal .slides section .fragment {
  opacity: 0;
  visibility: hidden;
  transition: all .2s ease; }
  .reveal .slides section .fragment.visible {
    opacity: 1;
    visibility: inherit; }

.reveal .slides section .fragment.grow {
  opacity: 1;
  visibility: inherit; }
  .reveal .slides section .fragment.grow.visible {
    -webkit-transform: scale(1.3);
            transform: scale(1.3); }

.reveal .slides section .fragment.shrink {
  opacity: 1;
  visibility: inherit; }
  .reveal .slides section .fragment.shrink.visible {
    -webkit-transform: scale(0.7);
            transform: scale(0.7); }

.reveal .slides section .fragment.zoom-in {
  -webkit-transform: scale(0.1);
          transform: scale(0.1); }
  .reveal .slides section .fragment.zoom-in.visible {
    -webkit-transform: none;
            transform: none; }

.reveal .slides section .fragment.fade-out {
  opacity: 1;
  visibility: inherit; }
  .reveal .slides section .fragment.fade-out.visible {
    opacity: 0;
    visibility: hidden; }

.reveal .slides section .fragment.semi-fade-out {
  opacity: 1;
  visibility: inherit; }
  .reveal .slides section .fragment.semi-fade-out.visible {
    opacity: 0.5;
    visibility: inherit; }

.reveal .slides section .fragment.strike {
  opacity: 1;
  visibility: inherit; }
  .reveal .slides section .fragment.strike.visible {
    text-decoration: line-through; }

.reveal .slides section .fragment.fade-up {
  -webkit-transform: translate(0, 20%);
          transform: translate(0, 20%); }
  .reveal .slides section .fragment.fade-up.visible {
    -webkit-transform: translate(0, 0);
            transform: translate(0, 0); }

.reveal .slides section .fragment.fade-down {
  -webkit-transform: translate(0, -20%);
          transform: translate(0, -20%); }
  .reveal .slides section .fragment.fade-down.visible {
    -webkit-transform: translate(0, 0);
            transform: translate(0, 0); }

.reveal .slides section .fragment.fade-right {
  -webkit-transform: translate(-20%, 0);
          transform: translate(-20%, 0); }
  .reveal .slides section .fragment.fade-right.visible {
    -webkit-transform: translate(0, 0);
            transform: translate(0, 0); }

.reveal .slides section .fragment.fade-left {
  -webkit-transform: translate(20%, 0);
          transform: translate(20%, 0); }
  .reveal .slides section .fragment.fade-left.visible {
    -webkit-transform: translate(0, 0);
            transform: translate(0, 0); }

.reveal .slides section .fragment.fade-in-then-out,
.reveal .slides section .fragment.current-visible {
  opacity: 0;
  visibility: hidden; }
  .reveal .slides section .fragment.fade-in-then-out.current-fragment,
  .reveal .slides section .fragment.current-visible.current-fragment {
    opacity: 1;
    visibility: inherit; }

.reveal .slides section .fragment.fade-in-then-semi-out {
  opacity: 0;
  visibility: hidden; }
  .reveal .slides section .fragment.fade-in-then-semi-out.visible {
    opacity: 0.5;
    visibility: inherit; }
  .reveal .slides section .fragment.fade-in-then-semi-out.current-fragment {
    opacity: 1;
    visibility: inherit; }

.reveal .slides section .fragment.highlight-red,
.reveal .slides section .fragment.highlight-current-red,
.reveal .slides section .fragment.highlight-green,
.reveal .slides section .fragment.highlight-current-green,
.reveal .slides section .fragment.highlight-blue,
.reveal .slides section .fragment.highlight-current-blue {
  opacity: 1;
  visibility: inherit; }

.reveal .slides section .fragment.highlight-red.visible {
  color: #ff2c2d; }

.reveal .slides section .fragment.highlight-green.visible {
  color: #17ff2e; }

.reveal .slides section .fragment.highlight-blue.visible {
  color: #1b91ff; }

.reveal .slides section .fragment.highlight-current-red.current-fragment {
  color: #ff2c2d; }

.reveal .slides section .fragment.highlight-current-green.current-fragment {
  color: #17ff2e; }

.reveal .slides section .fragment.highlight-current-blue.current-fragment {
  color: #1b91ff; }

/*********************************************
 * DEFAULT ELEMENT STYLES
 *********************************************/
/* Fixes issue in Chrome where italic fonts did not appear when printing to PDF */
.reveal:after {
  content: '';
  font-style: italic; }

.reveal iframe {
  z-index: 1; }

/** Prevents layering issues in certain browser/transition combinations */
.reveal a {
  position: relative; }

.reveal .stretch {
  max-width: none;
  max-height: none; }

.reveal pre.stretch code {
  height: 100%;
  max-height: 100%;
  box-sizing: border-box; }

/*********************************************
 * CONTROLS
 *********************************************/
@-webkit-keyframes bounce-right {
  0%, 10%, 25%, 40%, 50% {
    -webkit-transform: translateX(0);
            transform: translateX(0); }
  20% {
    -webkit-transform: translateX(10px);
            transform: translateX(10px); }
  30% {
    -webkit-transform: translateX(-5px);
            transform: translateX(-5px); } }
@keyframes bounce-right {
  0%, 10%, 25%, 40%, 50% {
    -webkit-transform: translateX(0);
            transform: translateX(0); }
  20% {
    -webkit-transform: translateX(10px);
            transform: translateX(10px); }
  30% {
    -webkit-transform: translateX(-5px);
            transform: translateX(-5px); } }

@-webkit-keyframes bounce-down {
  0%, 10%, 25%, 40%, 50% {
    -webkit-transform: translateY(0);
            transform: translateY(0); }
  20% {
    -webkit-transform: translateY(10px);
            transform: translateY(10px); }
  30% {
    -webkit-transform: translateY(-5px);
            transform: translateY(-5px); } }

@keyframes bounce-down {
  0%, 10%, 25%, 40%, 50% {
    -webkit-transform: translateY(0);
            transform: translateY(0); }
  20% {
    -webkit-transform: translateY(10px);
            transform: translateY(10px); }
  30% {
    -webkit-transform: translateY(-5px);
            transform: translateY(-5px); } }

.reveal .controls {
  display: none;
  position: absolute;
  top: auto;
  bottom: 12px;
  right: 12px;
  left: auto;
  z-index: 1;
  color: #000;
  pointer-events: none;
  font-size: 10px; }
  .reveal .controls button {
    position: absolute;
    padding: 0;
    background-color: transparent;
    border: 0;
    outline: 0;
    cursor: pointer;
    color: currentColor;
    -webkit-transform: scale(0.9999);
            transform: scale(0.9999);
    transition: color 0.2s ease, opacity 0.2s ease, -webkit-transform 0.2s ease;
    transition: color 0.2s ease, opacity 0.2s ease, transform 0.2s ease;
    z-index: 2;
    pointer-events: auto;
    font-size: inherit;
    visibility: hidden;
    opacity: 0;
    -webkit-appearance: none;
    -webkit-tap-highlight-color: transparent; }
  .reveal .controls .controls-arrow:before,
  .reveal .controls .controls-arrow:after {
    content: '';
    position: absolute;
    top: 0;
    left: 0;
    width: 2.6em;
    height: 0.5em;
    border-radius: 0.25em;
    background-color: currentColor;
    transition: all 0.15s ease, background-color 0.8s ease;
    -webkit-transform-origin: 0.2em 50%;
            transform-origin: 0.2em 50%;
    will-change: transform; }
  .reveal .controls .controls-arrow {
    position: relative;
    width: 3.6em;
    height: 3.6em; }
    .reveal .controls .controls-arrow:before {
      -webkit-transform: translateX(0.5em) translateY(1.55em) rotate(45deg);
              transform: translateX(0.5em) translateY(1.55em) rotate(45deg); }
    .reveal .controls .controls-arrow:after {
      -webkit-transform: translateX(0.5em) translateY(1.55em) rotate(-45deg);
              transform: translateX(0.5em) translateY(1.55em) rotate(-45deg); }
    .reveal .controls .controls-arrow:hover:before {
      -webkit-transform: translateX(0.5em) translateY(1.55em) rotate(40deg);
              transform: translateX(0.5em) translateY(1.55em) rotate(40deg); }
    .reveal .controls .controls-arrow:hover:after {
      -webkit-transform: translateX(0.5em) translateY(1.55em) rotate(-40deg);
              transform: translateX(0.5em) translateY(1.55em) rotate(-40deg); }
    .reveal .controls .controls-arrow:active:before {
      -webkit-transform: translateX(0.5em) translateY(1.55em) rotate(36deg);
              transform: translateX(0.5em) translateY(1.55em) rotate(36deg); }
    .reveal .controls .controls-arrow:active:after {
      -webkit-transform: translateX(0.5em) translateY(1.55em) rotate(-36deg);
              transform: translateX(0.5em) translateY(1.55em) rotate(-36deg); }
  .reveal .controls .navigate-left {
    right: 6.4em;
    bottom: 3.2em;
    -webkit-transform: translateX(-10px);
            transform: translateX(-10px); }
  .reveal .controls .navigate-right {
    right: 0;
    bottom: 3.2em;
    -webkit-transform: translateX(10px);
            transform: translateX(10px); }
    .reveal .controls .navigate-right .controls-arrow {
      -webkit-transform: rotate(180deg);
              transform: rotate(180deg); }
    .reveal .controls .navigate-right.highlight {
      -webkit-animation: bounce-right 2s 50 both ease-out;
              animation: bounce-right 2s 50 both ease-out; }
  .reveal .controls .navigate-up {
    right: 3.2em;
    bottom: 6.4em;
    -webkit-transform: translateY(-10px);
            transform: translateY(-10px); }
    .reveal .controls .navigate-up .controls-arrow {
      -webkit-transform: rotate(90deg);
              transform: rotate(90deg); }
  .reveal .controls .navigate-down {
    right: 3.2em;
    bottom: 0;
    -webkit-transform: translateY(10px);
            transform: translateY(10px); }
    .reveal .controls .navigate-down .controls-arrow {
      -webkit-transform: rotate(-90deg);
              transform: rotate(-90deg); }
    .reveal .controls .navigate-down.highlight {
      -webkit-animation: bounce-down 2s 50 both ease-out;
              animation: bounce-down 2s 50 both ease-out; }
  .reveal .controls[data-controls-back-arrows="faded"] .navigate-left.enabled,
  .reveal .controls[data-controls-back-arrows="faded"] .navigate-up.enabled {
    opacity: 0.3; }
    .reveal .controls[data-controls-back-arrows="faded"] .navigate-left.enabled:hover,
    .reveal .controls[data-controls-back-arrows="faded"] .navigate-up.enabled:hover {
      opacity: 1; }
  .reveal .controls[data-controls-back-arrows="hidden"] .navigate-left.enabled,
  .reveal .controls[data-controls-back-arrows="hidden"] .navigate-up.enabled {
    opacity: 0;
    visibility: hidden; }
  .reveal .controls .enabled {
    visibility: visible;
    opacity: 0.9;
    cursor: pointer;
    -webkit-transform: none;
            transform: none; }
  .reveal .controls .enabled.fragmented {
    opacity: 0.5; }
  .reveal .controls .enabled:hover,
  .reveal .controls .enabled.fragmented:hover {
    opacity: 1; }

.reveal:not(.has-vertical-slides) .controls .navigate-left {
  bottom: 1.4em;
  right: 5.5em; }

.reveal:not(.has-vertical-slides) .controls .navigate-right {
  bottom: 1.4em;
  right: 0.5em; }

.reveal:not(.has-horizontal-slides) .controls .navigate-up {
  right: 1.4em;
  bottom: 5em; }

.reveal:not(.has-horizontal-slides) .controls .navigate-down {
  right: 1.4em;
  bottom: 0.5em; }

.reveal.has-dark-background .controls {
  color: #fff; }

.reveal.has-light-background .controls {
  color: #000; }

.reveal.no-hover .controls .controls-arrow:hover:before,
.reveal.no-hover .controls .controls-arrow:active:before {
  -webkit-transform: translateX(0.5em) translateY(1.55em) rotate(45deg);
          transform: translateX(0.5em) translateY(1.55em) rotate(45deg); }

.reveal.no-hover .controls .controls-arrow:hover:after,
.reveal.no-hover .controls .controls-arrow:active:after {
  -webkit-transform: translateX(0.5em) translateY(1.55em) rotate(-45deg);
          transform: translateX(0.5em) translateY(1.55em) rotate(-45deg); }

@media screen and (min-width: 500px) {
  .reveal .controls[data-controls-layout="edges"] {
    top: 0;
    right: 0;
    bottom: 0;
    left: 0; }
  .reveal .controls[data-controls-layout="edges"] .navigate-left,
  .reveal .controls[data-controls-layout="edges"] .navigate-right,
  .reveal .controls[data-controls-layout="edges"] .navigate-up,
  .reveal .controls[data-controls-layout="edges"] .navigate-down {
    bottom: auto;
    right: auto; }
  .reveal .controls[data-controls-layout="edges"] .navigate-left {
    top: 50%;
    left: 8px;
    margin-top: -1.8em; }
  .reveal .controls[data-controls-layout="edges"] .navigate-right {
    top: 50%;
    right: 8px;
    margin-top: -1.8em; }
  .reveal .controls[data-controls-layout="edges"] .navigate-up {
    top: 8px;
    left: 50%;
    margin-left: -1.8em; }
  .reveal .controls[data-controls-layout="edges"] .navigate-down {
    bottom: 8px;
    left: 50%;
    margin-left: -1.8em; } }

/*********************************************
 * PROGRESS BAR
 *********************************************/
.reveal .progress {
  position: absolute;
  display: none;
  height: 3px;
  width: 100%;
  bottom: 0;
  left: 0;
  z-index: 10;
  background-color: rgba(0, 0, 0, 0.2);
  color: #fff; }

.reveal .progress:after {
  content: '';
  display: block;
  position: absolute;
  height: 10px;
  width: 100%;
  top: -10px; }

.reveal .progress span {
  display: block;
  height: 100%;
  width: 0px;
  background-color: currentColor;
  transition: width 800ms cubic-bezier(0.26, 0.86, 0.44, 0.985); }

/*********************************************
 * SLIDE NUMBER
 *********************************************/
.reveal .slide-number {
  position: absolute;
  display: block;
  right: 8px;
  bottom: 8px;
  z-index: 31;
  font-family: Helvetica, sans-serif;
  font-size: 12px;
  line-height: 1;
  color: #fff;
  background-color: rgba(0, 0, 0, 0.4);
  padding: 5px; }

.reveal .slide-number a {
  color: currentColor; }

.reveal .slide-number-delimiter {
  margin: 0 3px; }

/*********************************************
 * SLIDES
 *********************************************/
.reveal {
  position: relative;
  width: 100%;
  height: 100%;
  overflow: hidden;
  -ms-touch-action: none;
      touch-action: none; }

@media only screen and (orientation: landscape) {
  .reveal.ua-iphone {
    position: fixed; } }

.reveal .slides {
  position: absolute;
  width: 100%;
  height: 100%;
  top: 0;
  right: 0;
  bottom: 0;
  left: 0;
  margin: auto;
  pointer-events: none;
  overflow: visible;
  z-index: 1;
  text-align: center;
  -webkit-perspective: 600px;
          perspective: 600px;
  -webkit-perspective-origin: 50% 40%;
          perspective-origin: 50% 40%; }

.reveal .slides > section {
  -ms-perspective: 600px; }

.reveal .slides > section,
.reveal .slides > section > section {
  display: none;
  position: absolute;
  width: 100%;
  padding: 20px 0px;
  pointer-events: auto;
  z-index: 10;
  -webkit-transform-style: flat;
          transform-style: flat;
  transition: -webkit-transform-origin 800ms cubic-bezier(0.26, 0.86, 0.44, 0.985), -webkit-transform 800ms cubic-bezier(0.26, 0.86, 0.44, 0.985), visibility 800ms cubic-bezier(0.26, 0.86, 0.44, 0.985), opacity 800ms cubic-bezier(0.26, 0.86, 0.44, 0.985);
  transition: transform-origin 800ms cubic-bezier(0.26, 0.86, 0.44, 0.985), transform 800ms cubic-bezier(0.26, 0.86, 0.44, 0.985), visibility 800ms cubic-bezier(0.26, 0.86, 0.44, 0.985), opacity 800ms cubic-bezier(0.26, 0.86, 0.44, 0.985); }

/* Global transition speed settings */
.reveal[data-transition-speed="fast"] .slides section {
  transition-duration: 400ms; }

.reveal[data-transition-speed="slow"] .slides section {
  transition-duration: 1200ms; }

/* Slide-specific transition speed overrides */
.reveal .slides section[data-transition-speed="fast"] {
  transition-duration: 400ms; }

.reveal .slides section[data-transition-speed="slow"] {
  transition-duration: 1200ms; }

.reveal .slides > section.stack {
  padding-top: 0;
  padding-bottom: 0;
  pointer-events: none; }

.reveal .slides > section.present,
.reveal .slides > section > section.present {
  display: block;
  z-index: 11;
  opacity: 1; }

.reveal .slides > section:empty,
.reveal .slides > section > section:empty,
.reveal .slides > section[data-background-interactive],
.reveal .slides > section > section[data-background-interactive] {
  pointer-events: none; }

.reveal.center,
.reveal.center .slides,
.reveal.center .slides section {
  min-height: 0 !important; }

/* Don't allow interaction with invisible slides */
.reveal .slides > section.future,
.reveal .slides > section > section.future,
.reveal .slides > section.past,
.reveal .slides > section > section.past {
  pointer-events: none; }

.reveal.overview .slides > section,
.reveal.overview .slides > section > section {
  pointer-events: auto; }

.reveal .slides > section.past,
.reveal .slides > section.future,
.reveal .slides > section > section.past,
.reveal .slides > section > section.future {
  opacity: 0; }

/*********************************************
 * Mixins for readability of transitions
 *********************************************/
/*********************************************
 * SLIDE TRANSITION
 * Aliased 'linear' for backwards compatibility
 *********************************************/
.reveal.slide section {
  -webkit-backface-visibility: hidden;
          backface-visibility: hidden; }

.reveal .slides > section[data-transition=slide].past,
.reveal .slides > section[data-transition~=slide-out].past,
.reveal.slide .slides > section:not([data-transition]).past {
  -webkit-transform: translate(-150%, 0);
          transform: translate(-150%, 0); }

.reveal .slides > section[data-transition=slide].future,
.reveal .slides > section[data-transition~=slide-in].future,
.reveal.slide .slides > section:not([data-transition]).future {
  -webkit-transform: translate(150%, 0);
          transform: translate(150%, 0); }

.reveal .slides > section > section[data-transition=slide].past,
.reveal .slides > section > section[data-transition~=slide-out].past,
.reveal.slide .slides > section > section:not([data-transition]).past {
  -webkit-transform: translate(0, -150%);
          transform: translate(0, -150%); }

.reveal .slides > section > section[data-transition=slide].future,
.reveal .slides > section > section[data-transition~=slide-in].future,
.reveal.slide .slides > section > section:not([data-transition]).future {
  -webkit-transform: translate(0, 150%);
          transform: translate(0, 150%); }

.reveal.linear section {
  -webkit-backface-visibility: hidden;
          backface-visibility: hidden; }

.reveal .slides > section[data-transition=linear].past,
.reveal .slides > section[data-transition~=linear-out].past,
.reveal.linear .slides > section:not([data-transition]).past {
  -webkit-transform: translate(-150%, 0);
          transform: translate(-150%, 0); }

.reveal .slides > section[data-transition=linear].future,
.reveal .slides > section[data-transition~=linear-in].future,
.reveal.linear .slides > section:not([data-transition]).future {
  -webkit-transform: translate(150%, 0);
          transform: translate(150%, 0); }

.reveal .slides > section > section[data-transition=linear].past,
.reveal .slides > section > section[data-transition~=linear-out].past,
.reveal.linear .slides > section > section:not([data-transition]).past {
  -webkit-transform: translate(0, -150%);
          transform: translate(0, -150%); }

.reveal .slides > section > section[data-transition=linear].future,
.reveal .slides > section > section[data-transition~=linear-in].future,
.reveal.linear .slides > section > section:not([data-transition]).future {
  -webkit-transform: translate(0, 150%);
          transform: translate(0, 150%); }

/*********************************************
 * CONVEX TRANSITION
 * Aliased 'default' for backwards compatibility
 *********************************************/
.reveal .slides section[data-transition=default].stack,
.reveal.default .slides section.stack {
  -webkit-transform-style: preserve-3d;
          transform-style: preserve-3d; }

.reveal .slides > section[data-transition=default].past,
.reveal .slides > section[data-transition~=default-out].past,
.reveal.default .slides > section:not([data-transition]).past {
  -webkit-transform: translate3d(-100%, 0, 0) rotateY(-90deg) translate3d(-100%, 0, 0);
          transform: translate3d(-100%, 0, 0) rotateY(-90deg) translate3d(-100%, 0, 0); }

.reveal .slides > section[data-transition=default].future,
.reveal .slides > section[data-transition~=default-in].future,
.reveal.default .slides > section:not([data-transition]).future {
  -webkit-transform: translate3d(100%, 0, 0) rotateY(90deg) translate3d(100%, 0, 0);
          transform: translate3d(100%, 0, 0) rotateY(90deg) translate3d(100%, 0, 0); }

.reveal .slides > section > section[data-transition=default].past,
.reveal .slides > section > section[data-transition~=default-out].past,
.reveal.default .slides > section > section:not([data-transition]).past {
  -webkit-transform: translate3d(0, -300px, 0) rotateX(70deg) translate3d(0, -300px, 0);
          transform: translate3d(0, -300px, 0) rotateX(70deg) translate3d(0, -300px, 0); }

.reveal .slides > section > section[data-transition=default].future,
.reveal .slides > section > section[data-transition~=default-in].future,
.reveal.default .slides > section > section:not([data-transition]).future {
  -webkit-transform: translate3d(0, 300px, 0) rotateX(-70deg) translate3d(0, 300px, 0);
          transform: translate3d(0, 300px, 0) rotateX(-70deg) translate3d(0, 300px, 0); }

.reveal .slides section[data-transition=convex].stack,
.reveal.convex .slides section.stack {
  -webkit-transform-style: preserve-3d;
          transform-style: preserve-3d; }

.reveal .slides > section[data-transition=convex].past,
.reveal .slides > section[data-transition~=convex-out].past,
.reveal.convex .slides > section:not([data-transition]).past {
  -webkit-transform: translate3d(-100%, 0, 0) rotateY(-90deg) translate3d(-100%, 0, 0);
          transform: translate3d(-100%, 0, 0) rotateY(-90deg) translate3d(-100%, 0, 0); }

.reveal .slides > section[data-transition=convex].future,
.reveal .slides > section[data-transition~=convex-in].future,
.reveal.convex .slides > section:not([data-transition]).future {
  -webkit-transform: translate3d(100%, 0, 0) rotateY(90deg) translate3d(100%, 0, 0);
          transform: translate3d(100%, 0, 0) rotateY(90deg) translate3d(100%, 0, 0); }

.reveal .slides > section > section[data-transition=convex].past,
.reveal .slides > section > section[data-transition~=convex-out].past,
.reveal.convex .slides > section > section:not([data-transition]).past {
  -webkit-transform: translate3d(0, -300px, 0) rotateX(70deg) translate3d(0, -300px, 0);
          transform: translate3d(0, -300px, 0) rotateX(70deg) translate3d(0, -300px, 0); }

.reveal .slides > section > section[data-transition=convex].future,
.reveal .slides > section > section[data-transition~=convex-in].future,
.reveal.convex .slides > section > section:not([data-transition]).future {
  -webkit-transform: translate3d(0, 300px, 0) rotateX(-70deg) translate3d(0, 300px, 0);
          transform: translate3d(0, 300px, 0) rotateX(-70deg) translate3d(0, 300px, 0); }

/*********************************************
 * CONCAVE TRANSITION
 *********************************************/
.reveal .slides section[data-transition=concave].stack,
.reveal.concave .slides section.stack {
  -webkit-transform-style: preserve-3d;
          transform-style: preserve-3d; }

.reveal .slides > section[data-transition=concave].past,
.reveal .slides > section[data-transition~=concave-out].past,
.reveal.concave .slides > section:not([data-transition]).past {
  -webkit-transform: translate3d(-100%, 0, 0) rotateY(90deg) translate3d(-100%, 0, 0);
          transform: translate3d(-100%, 0, 0) rotateY(90deg) translate3d(-100%, 0, 0); }

.reveal .slides > section[data-transition=concave].future,
.reveal .slides > section[data-transition~=concave-in].future,
.reveal.concave .slides > section:not([data-transition]).future {
  -webkit-transform: translate3d(100%, 0, 0) rotateY(-90deg) translate3d(100%, 0, 0);
          transform: translate3d(100%, 0, 0) rotateY(-90deg) translate3d(100%, 0, 0); }

.reveal .slides > section > section[data-transition=concave].past,
.reveal .slides > section > section[data-transition~=concave-out].past,
.reveal.concave .slides > section > section:not([data-transition]).past {
  -webkit-transform: translate3d(0, -80%, 0) rotateX(-70deg) translate3d(0, -80%, 0);
          transform: translate3d(0, -80%, 0) rotateX(-70deg) translate3d(0, -80%, 0); }

.reveal .slides > section > section[data-transition=concave].future,
.reveal .slides > section > section[data-transition~=concave-in].future,
.reveal.concave .slides > section > section:not([data-transition]).future {
  -webkit-transform: translate3d(0, 80%, 0) rotateX(70deg) translate3d(0, 80%, 0);
          transform: translate3d(0, 80%, 0) rotateX(70deg) translate3d(0, 80%, 0); }

/*********************************************
 * ZOOM TRANSITION
 *********************************************/
.reveal .slides section[data-transition=zoom],
.reveal.zoom .slides section:not([data-transition]) {
  transition-timing-function: ease; }

.reveal .slides > section[data-transition=zoom].past,
.reveal .slides > section[data-transition~=zoom-out].past,
.reveal.zoom .slides > section:not([data-transition]).past {
  visibility: hidden;
  -webkit-transform: scale(16);
          transform: scale(16); }

.reveal .slides > section[data-transition=zoom].future,
.reveal .slides > section[data-transition~=zoom-in].future,
.reveal.zoom .slides > section:not([data-transition]).future {
  visibility: hidden;
  -webkit-transform: scale(0.2);
          transform: scale(0.2); }

.reveal .slides > section > section[data-transition=zoom].past,
.reveal .slides > section > section[data-transition~=zoom-out].past,
.reveal.zoom .slides > section > section:not([data-transition]).past {
  -webkit-transform: translate(0, -150%);
          transform: translate(0, -150%); }

.reveal .slides > section > section[data-transition=zoom].future,
.reveal .slides > section > section[data-transition~=zoom-in].future,
.reveal.zoom .slides > section > section:not([data-transition]).future {
  -webkit-transform: translate(0, 150%);
          transform: translate(0, 150%); }

/*********************************************
 * CUBE TRANSITION
 *
 * WARNING:
 * this is deprecated and will be removed in a
 * future version.
 *********************************************/
.reveal.cube .slides {
  -webkit-perspective: 1300px;
          perspective: 1300px; }

.reveal.cube .slides section {
  padding: 30px;
  min-height: 700px;
  -webkit-backface-visibility: hidden;
          backface-visibility: hidden;
  box-sizing: border-box;
  -webkit-transform-style: preserve-3d;
          transform-style: preserve-3d; }

.reveal.center.cube .slides section {
  min-height: 0; }

.reveal.cube .slides section:not(.stack):before {
  content: '';
  position: absolute;
  display: block;
  width: 100%;
  height: 100%;
  left: 0;
  top: 0;
  background: rgba(0, 0, 0, 0.1);
  border-radius: 4px;
  -webkit-transform: translateZ(-20px);
          transform: translateZ(-20px); }

.reveal.cube .slides section:not(.stack):after {
  content: '';
  position: absolute;
  display: block;
  width: 90%;
  height: 30px;
  left: 5%;
  bottom: 0;
  background: none;
  z-index: 1;
  border-radius: 4px;
  box-shadow: 0px 95px 25px rgba(0, 0, 0, 0.2);
  -webkit-transform: translateZ(-90px) rotateX(65deg);
          transform: translateZ(-90px) rotateX(65deg); }

.reveal.cube .slides > section.stack {
  padding: 0;
  background: none; }

.reveal.cube .slides > section.past {
  -webkit-transform-origin: 100% 0%;
          transform-origin: 100% 0%;
  -webkit-transform: translate3d(-100%, 0, 0) rotateY(-90deg);
          transform: translate3d(-100%, 0, 0) rotateY(-90deg); }

.reveal.cube .slides > section.future {
  -webkit-transform-origin: 0% 0%;
          transform-origin: 0% 0%;
  -webkit-transform: translate3d(100%, 0, 0) rotateY(90deg);
          transform: translate3d(100%, 0, 0) rotateY(90deg); }

.reveal.cube .slides > section > section.past {
  -webkit-transform-origin: 0% 100%;
          transform-origin: 0% 100%;
  -webkit-transform: translate3d(0, -100%, 0) rotateX(90deg);
          transform: translate3d(0, -100%, 0) rotateX(90deg); }

.reveal.cube .slides > section > section.future {
  -webkit-transform-origin: 0% 0%;
          transform-origin: 0% 0%;
  -webkit-transform: translate3d(0, 100%, 0) rotateX(-90deg);
          transform: translate3d(0, 100%, 0) rotateX(-90deg); }

/*********************************************
 * PAGE TRANSITION
 *
 * WARNING:
 * this is deprecated and will be removed in a
 * future version.
 *********************************************/
.reveal.page .slides {
  -webkit-perspective-origin: 0% 50%;
          perspective-origin: 0% 50%;
  -webkit-perspective: 3000px;
          perspective: 3000px; }

.reveal.page .slides section {
  padding: 30px;
  min-height: 700px;
  box-sizing: border-box;
  -webkit-transform-style: preserve-3d;
          transform-style: preserve-3d; }

.reveal.page .slides section.past {
  z-index: 12; }

.reveal.page .slides section:not(.stack):before {
  content: '';
  position: absolute;
  display: block;
  width: 100%;
  height: 100%;
  left: 0;
  top: 0;
  background: rgba(0, 0, 0, 0.1);
  -webkit-transform: translateZ(-20px);
          transform: translateZ(-20px); }

.reveal.page .slides section:not(.stack):after {
  content: '';
  position: absolute;
  display: block;
  width: 90%;
  height: 30px;
  left: 5%;
  bottom: 0;
  background: none;
  z-index: 1;
  border-radius: 4px;
  box-shadow: 0px 95px 25px rgba(0, 0, 0, 0.2);
  -webkit-transform: translateZ(-90px) rotateX(65deg); }

.reveal.page .slides > section.stack {
  padding: 0;
  background: none; }

.reveal.page .slides > section.past {
  -webkit-transform-origin: 0% 0%;
          transform-origin: 0% 0%;
  -webkit-transform: translate3d(-40%, 0, 0) rotateY(-80deg);
          transform: translate3d(-40%, 0, 0) rotateY(-80deg); }

.reveal.page .slides > section.future {
  -webkit-transform-origin: 100% 0%;
          transform-origin: 100% 0%;
  -webkit-transform: translate3d(0, 0, 0);
          transform: translate3d(0, 0, 0); }

.reveal.page .slides > section > section.past {
  -webkit-transform-origin: 0% 0%;
          transform-origin: 0% 0%;
  -webkit-transform: translate3d(0, -40%, 0) rotateX(80deg);
          transform: translate3d(0, -40%, 0) rotateX(80deg); }

.reveal.page .slides > section > section.future {
  -webkit-transform-origin: 0% 100%;
          transform-origin: 0% 100%;
  -webkit-transform: translate3d(0, 0, 0);
          transform: translate3d(0, 0, 0); }

/*********************************************
 * FADE TRANSITION
 *********************************************/
.reveal .slides section[data-transition=fade],
.reveal.fade .slides section:not([data-transition]),
.reveal.fade .slides > section > section:not([data-transition]) {
  -webkit-transform: none;
          transform: none;
  transition: opacity 0.5s; }

.reveal.fade.overview .slides section,
.reveal.fade.overview .slides > section > section {
  transition: none; }

/*********************************************
 * NO TRANSITION
 *********************************************/
.reveal .slides section[data-transition=none],
.reveal.none .slides section:not([data-transition]) {
  -webkit-transform: none;
          transform: none;
  transition: none; }

/*********************************************
 * PAUSED MODE
 *********************************************/
.reveal .pause-overlay {
  position: absolute;
  top: 0;
  left: 0;
  width: 100%;
  height: 100%;
  background: black;
  visibility: hidden;
  opacity: 0;
  z-index: 100;
  transition: all 1s ease; }

.reveal .pause-overlay .resume-button {
  position: absolute;
  bottom: 20px;
  right: 20px;
  color: #ccc;
  border-radius: 2px;
  padding: 6px 14px;
  border: 2px solid #ccc;
  font-size: 16px;
  background: transparent;
  cursor: pointer; }
  .reveal .pause-overlay .resume-button:hover {
    color: #fff;
    border-color: #fff; }

.reveal.paused .pause-overlay {
  visibility: visible;
  opacity: 1; }

/*********************************************
 * FALLBACK
 *********************************************/
.no-transforms {
  overflow-y: auto; }

.no-transforms .reveal .slides {
  position: relative;
  width: 80%;
  height: auto !important;
  top: 0;
  left: 50%;
  margin: 0;
  text-align: center; }

.no-transforms .reveal .controls,
.no-transforms .reveal .progress {
  display: none !important; }

.no-transforms .reveal .slides section {
  display: block !important;
  opacity: 1 !important;
  position: relative !important;
  height: auto;
  min-height: 0;
  top: 0;
  left: -50%;
  margin: 70px 0;
  -webkit-transform: none;
          transform: none; }

.no-transforms .reveal .slides section section {
  left: 0; }

.reveal .no-transition,
.reveal .no-transition * {
  transition: none !important; }

/*********************************************
 * PER-SLIDE BACKGROUNDS
 *********************************************/
.reveal .backgrounds {
  position: absolute;
  width: 100%;
  height: 100%;
  top: 0;
  left: 0;
  -webkit-perspective: 600px;
          perspective: 600px; }

.reveal .slide-background {
  display: none;
  position: absolute;
  width: 100%;
  height: 100%;
  opacity: 0;
  visibility: hidden;
  overflow: hidden;
  background-color: transparent;
  transition: all 800ms cubic-bezier(0.26, 0.86, 0.44, 0.985); }

.reveal .slide-background-content {
  position: absolute;
  width: 100%;
  height: 100%;
  background-position: 50% 50%;
  background-repeat: no-repeat;
  background-size: cover; }

.reveal .slide-background.stack {
  display: block; }

.reveal .slide-background.present {
  opacity: 1;
  visibility: visible;
  z-index: 2; }

.print-pdf .reveal .slide-background {
  opacity: 1 !important;
  visibility: visible !important; }

/* Video backgrounds */
.reveal .slide-background video {
  position: absolute;
  width: 100%;
  height: 100%;
  max-width: none;
  max-height: none;
  top: 0;
  left: 0;
  -o-object-fit: cover;
     object-fit: cover; }

.reveal .slide-background[data-background-size="contain"] video {
  -o-object-fit: contain;
     object-fit: contain; }

/* Immediate transition style */
.reveal[data-background-transition=none] > .backgrounds .slide-background,
.reveal > .backgrounds .slide-background[data-background-transition=none] {
  transition: none; }

/* Slide */
.reveal[data-background-transition=slide] > .backgrounds .slide-background,
.reveal > .backgrounds .slide-background[data-background-transition=slide] {
  opacity: 1;
  -webkit-backface-visibility: hidden;
          backface-visibility: hidden; }

.reveal[data-background-transition=slide] > .backgrounds .slide-background.past,
.reveal > .backgrounds .slide-background.past[data-background-transition=slide] {
  -webkit-transform: translate(-100%, 0);
          transform: translate(-100%, 0); }

.reveal[data-background-transition=slide] > .backgrounds .slide-background.future,
.reveal > .backgrounds .slide-background.future[data-background-transition=slide] {
  -webkit-transform: translate(100%, 0);
          transform: translate(100%, 0); }

.reveal[data-background-transition=slide] > .backgrounds .slide-background > .slide-background.past,
.reveal > .backgrounds .slide-background > .slide-background.past[data-background-transition=slide] {
  -webkit-transform: translate(0, -100%);
          transform: translate(0, -100%); }

.reveal[data-background-transition=slide] > .backgrounds .slide-background > .slide-background.future,
.reveal > .backgrounds .slide-background > .slide-background.future[data-background-transition=slide] {
  -webkit-transform: translate(0, 100%);
          transform: translate(0, 100%); }

/* Convex */
.reveal[data-background-transition=convex] > .backgrounds .slide-background.past,
.reveal > .backgrounds .slide-background.past[data-background-transition=convex] {
  opacity: 0;
  -webkit-transform: translate3d(-100%, 0, 0) rotateY(-90deg) translate3d(-100%, 0, 0);
          transform: translate3d(-100%, 0, 0) rotateY(-90deg) translate3d(-100%, 0, 0); }

.reveal[data-background-transition=convex] > .backgrounds .slide-background.future,
.reveal > .backgrounds .slide-background.future[data-background-transition=convex] {
  opacity: 0;
  -webkit-transform: translate3d(100%, 0, 0) rotateY(90deg) translate3d(100%, 0, 0);
          transform: translate3d(100%, 0, 0) rotateY(90deg) translate3d(100%, 0, 0); }

.reveal[data-background-transition=convex] > .backgrounds .slide-background > .slide-background.past,
.reveal > .backgrounds .slide-background > .slide-background.past[data-background-transition=convex] {
  opacity: 0;
  -webkit-transform: translate3d(0, -100%, 0) rotateX(90deg) translate3d(0, -100%, 0);
          transform: translate3d(0, -100%, 0) rotateX(90deg) translate3d(0, -100%, 0); }

.reveal[data-background-transition=convex] > .backgrounds .slide-background > .slide-background.future,
.reveal > .backgrounds .slide-background > .slide-background.future[data-background-transition=convex] {
  opacity: 0;
  -webkit-transform: translate3d(0, 100%, 0) rotateX(-90deg) translate3d(0, 100%, 0);
          transform: translate3d(0, 100%, 0) rotateX(-90deg) translate3d(0, 100%, 0); }

/* Concave */
.reveal[data-background-transition=concave] > .backgrounds .slide-background.past,
.reveal > .backgrounds .slide-background.past[data-background-transition=concave] {
  opacity: 0;
  -webkit-transform: translate3d(-100%, 0, 0) rotateY(90deg) translate3d(-100%, 0, 0);
          transform: translate3d(-100%, 0, 0) rotateY(90deg) translate3d(-100%, 0, 0); }

.reveal[data-background-transition=concave] > .backgrounds .slide-background.future,
.reveal > .backgrounds .slide-background.future[data-background-transition=concave] {
  opacity: 0;
  -webkit-transform: translate3d(100%, 0, 0) rotateY(-90deg) translate3d(100%, 0, 0);
          transform: translate3d(100%, 0, 0) rotateY(-90deg) translate3d(100%, 0, 0); }

.reveal[data-background-transition=concave] > .backgrounds .slide-background > .slide-background.past,
.reveal > .backgrounds .slide-background > .slide-background.past[data-background-transition=concave] {
  opacity: 0;
  -webkit-transform: translate3d(0, -100%, 0) rotateX(-90deg) translate3d(0, -100%, 0);
          transform: translate3d(0, -100%, 0) rotateX(-90deg) translate3d(0, -100%, 0); }

.reveal[data-background-transition=concave] > .backgrounds .slide-background > .slide-background.future,
.reveal > .backgrounds .slide-background > .slide-background.future[data-background-transition=concave] {
  opacity: 0;
  -webkit-transform: translate3d(0, 100%, 0) rotateX(90deg) translate3d(0, 100%, 0);
          transform: translate3d(0, 100%, 0) rotateX(90deg) translate3d(0, 100%, 0); }

/* Zoom */
.reveal[data-background-transition=zoom] > .backgrounds .slide-background,
.reveal > .backgrounds .slide-background[data-background-transition=zoom] {
  transition-timing-function: ease; }

.reveal[data-background-transition=zoom] > .backgrounds .slide-background.past,
.reveal > .backgrounds .slide-background.past[data-background-transition=zoom] {
  opacity: 0;
  visibility: hidden;
  -webkit-transform: scale(16);
          transform: scale(16); }

.reveal[data-background-transition=zoom] > .backgrounds .slide-background.future,
.reveal > .backgrounds .slide-background.future[data-background-transition=zoom] {
  opacity: 0;
  visibility: hidden;
  -webkit-transform: scale(0.2);
          transform: scale(0.2); }

.reveal[data-background-transition=zoom] > .backgrounds .slide-background > .slide-background.past,
.reveal > .backgrounds .slide-background > .slide-background.past[data-background-transition=zoom] {
  opacity: 0;
  visibility: hidden;
  -webkit-transform: scale(16);
          transform: scale(16); }

.reveal[data-background-transition=zoom] > .backgrounds .slide-background > .slide-background.future,
.reveal > .backgrounds .slide-background > .slide-background.future[data-background-transition=zoom] {
  opacity: 0;
  visibility: hidden;
  -webkit-transform: scale(0.2);
          transform: scale(0.2); }

/* Global transition speed settings */
.reveal[data-transition-speed="fast"] > .backgrounds .slide-background {
  transition-duration: 400ms; }

.reveal[data-transition-speed="slow"] > .backgrounds .slide-background {
  transition-duration: 1200ms; }

/*********************************************
 * OVERVIEW
 *********************************************/
.reveal.overview {
  -webkit-perspective-origin: 50% 50%;
          perspective-origin: 50% 50%;
  -webkit-perspective: 700px;
          perspective: 700px; }
  .reveal.overview .slides {
    -moz-transform-style: preserve-3d; }
  .reveal.overview .slides section {
    height: 100%;
    top: 0 !important;
    opacity: 1 !important;
    overflow: hidden;
    visibility: visible !important;
    cursor: pointer;
    box-sizing: border-box; }
  .reveal.overview .slides section:hover,
  .reveal.overview .slides section.present {
    outline: 10px solid rgba(150, 150, 150, 0.4);
    outline-offset: 10px; }
  .reveal.overview .slides section .fragment {
    opacity: 1;
    transition: none; }
  .reveal.overview .slides section:after,
  .reveal.overview .slides section:before {
    display: none !important; }
  .reveal.overview .slides > section.stack {
    padding: 0;
    top: 0 !important;
    background: none;
    outline: none;
    overflow: visible; }
  .reveal.overview .backgrounds {
    -webkit-perspective: inherit;
            perspective: inherit;
    -moz-transform-style: preserve-3d; }
  .reveal.overview .backgrounds .slide-background {
    opacity: 1;
    visibility: visible;
    outline: 10px solid rgba(150, 150, 150, 0.1);
    outline-offset: 10px; }
  .reveal.overview .backgrounds .slide-background.stack {
    overflow: visible; }

.reveal.overview .slides section,
.reveal.overview-deactivating .slides section {
  transition: none; }

.reveal.overview .backgrounds .slide-background,
.reveal.overview-deactivating .backgrounds .slide-background {
  transition: none; }

/*********************************************
 * RTL SUPPORT
 *********************************************/
.reveal.rtl .slides,
.reveal.rtl .slides h1,
.reveal.rtl .slides h2,
.reveal.rtl .slides h3,
.reveal.rtl .slides h4,
.reveal.rtl .slides h5,
.reveal.rtl .slides h6 {
  direction: rtl;
  font-family: sans-serif; }

.reveal.rtl pre,
.reveal.rtl code {
  direction: ltr; }

.reveal.rtl ol,
.reveal.rtl ul {
  text-align: right; }

.reveal.rtl .progress span {
  float: right; }

/*********************************************
 * PARALLAX BACKGROUND
 *********************************************/
.reveal.has-parallax-background .backgrounds {
  transition: all 0.8s ease; }

/* Global transition speed settings */
.reveal.has-parallax-background[data-transition-speed="fast"] .backgrounds {
  transition-duration: 400ms; }

.reveal.has-parallax-background[data-transition-speed="slow"] .backgrounds {
  transition-duration: 1200ms; }

/*********************************************
 * LINK PREVIEW OVERLAY
 *********************************************/
.reveal .overlay {
  position: absolute;
  top: 0;
  left: 0;
  width: 100%;
  height: 100%;
  z-index: 1000;
  background: rgba(0, 0, 0, 0.9);
  opacity: 0;
  visibility: hidden;
  transition: all 0.3s ease; }

.reveal .overlay.visible {
  opacity: 1;
  visibility: visible; }

.reveal .overlay .spinner {
  position: absolute;
  display: block;
  top: 50%;
  left: 50%;
  width: 32px;
  height: 32px;
  margin: -16px 0 0 -16px;
  z-index: 10;
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    <h1 style="width: 50%">
        &#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;<br> 
        <span style="font-size: 0.6em">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x8FDB;&#x9636;</span>
    </h1>
    <h2>&#x5F20;&#x817E;</h2>
    <h3>tengzhang@hust.edu.cn</h3>
</div>
</section><section><section data-notes lineno="25" class="slide " data-line="25" data-h="1" data-v="0">
<div class="multi_column">
    <div class="title_hr"> 
        <hr class="hr_top">
        <h5 class="title">&#x6838;&#x5FC3;&#x95EE;&#x9898;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p style="text-align:center;margin-top:20%">
    &#x5982;&#x4F55;&#x5C06;&#x5377;&#x79EF;&#x7684;&#x6982;&#x5FF5;&#x63A8;&#x5E7F;&#x5230;&#x56FE;&#x6570;&#x636E;&#xFF1F;
</p>
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        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x8FDB;&#x9636;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
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</section><section vertical="true" data-notes lineno="47" class="slide " data-line="47" data-h="1" data-v="1">
<div class="multi_column">
    <div class="title_hr"> 
        <hr class="hr_top">
        <h5 class="title">Fourier&#x53D8;&#x6362;</h5>
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    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
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<p>&#x51FD;&#x6570;<span class="mathjax-exps">$f$</span>&#x7684; Fourier &#x53D8;&#x6362;&#x4E3A;</p>
<div>
    $$
        \begin{align*}
            \Ffrak[f] = \fhat(w) = \int_{\Rbb} f(x) e^{-i w x} \diff x
        \end{align*}
    $$
</div>
<p>&#x4E8E;&#x662F;</p>
<div>
    $$
        \begin{align*}
            &amp; \lim_{N \rightarrow \infty} \int_{-N}^N \fhat(w) e^{i w y} \diff w = \lim_{N \rightarrow \infty} \int_{-N}^N \left[ \int_{\Rbb} f(x) e^{-i w x} \diff x \right]e^{i w y} \diff w \\
            &amp; = \int_{\Rbb} f(x) \left[ \lim_{N \rightarrow \infty} \left. \frac{e^{i w (y - x)}}{i(y-x)} \right|_{-N}^N \right] \diff x = \int_{\Rbb} f(x) \left[ \lim_{N \rightarrow \infty} \frac{ 2 i \sin (N (y - x))}{i(y-x)} \right] \diff x \\
            &amp; = 2 \pi \int_{\Rbb} f(y+z) \left[ \lim_{N \rightarrow \infty} \frac{\sin (N z)}{\pi z} \right] \diff z \\
            &amp; = 2 \pi \int_{\Rbb} f(y+z) \delta(z) \diff z = 2 \pi f(y)
        \end{align*}
    $$
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        <h6 class="bottom_center">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x8FDB;&#x9636;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="88" class="slide " data-line="88" data-h="1" data-v="2">
<div class="multi_column">
    <div class="title_hr"> 
        <hr class="hr_top">
        <h5 class="title">sinc&#x51FD;&#x6570;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<div>
    $$
        \begin{align*}
            \int_\Rbb \frac{\sin z}{z} \diff z &amp; = 2 \int_0^\infty \sin z \frac{1}{z} \diff z = 2 \int_0^\infty \sin z \left( \int_0^\infty e^{-zy} \diff y \right) \diff z
        \end{align*}
    $$
</div>
<p>&#x5176;&#x4E2D;</p>
<div>
    $$
        \begin{align*}
            \int_0^\infty &amp; e^{-yz} \sin z \diff z = \int_0^\infty e^{-yz} \frac{e^{iz} - e^{-iz}}{2 i } \diff z \\
            &amp; = \frac{1}{2 i} \int_0^\infty (e^{-(y - i)z} - e^{-(y+i)z}) \diff z = \frac{1}{2i} \left. \left( -\frac{e^{-(y - i)z}}{y - i} + \frac{e^{-(y + i)z}}{y + i} \right) \right|_0^\infty \\
            &amp; = \frac{1}{2i} \left( \frac{1}{y - i} - \frac{1}{y + i} \right) = \frac{1}{y^2 + 1} 
        \end{align*}
    $$
</div>
<p>&#x4E8E;&#x662F;</p>
<div>
    $$
        \begin{align*}
            \int_\Rbb \frac{\sin z}{z} \diff z &amp; = 2 \int_0^\infty \frac{1}{y^2 + 1} \diff y = 2 \arctan y |_0^\infty = \pi
        \end{align*}
    $$
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        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x8FDB;&#x9636;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="136" class="slide " data-line="136" data-h="1" data-v="3">
<div class="multi_column">
    <div class="title_hr"> 
        <hr class="hr_top">
        <h5 class="title">sinc&#x51FD;&#x6570;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<div>
    $$
        \begin{align*}
            \int_\Rbb \frac{\sin z}{\pi z} \diff z = 1 \Longrightarrow \int_\Rbb \frac{\sin N z}{\pi N z} \diff (N z) = 1 \Longrightarrow \int_\Rbb \frac{\sin N z}{\pi z} \diff z = 1
        \end{align*}
    $$
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        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x8FDB;&#x9636;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
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</section><section vertical="true" data-notes lineno="164" class="slide " data-line="164" data-h="1" data-v="4">
<div class="multi_column">
    <div class="title_hr"> 
        <hr class="hr_top">
        <h5 class="title">Fourier&#x53D8;&#x6362;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x51FD;&#x6570;<span class="mathjax-exps">$f$</span>&#x7684; Fourier &#x53D8;&#x6362;&#x4E3A;</p>
<div>
    $$
        \begin{align*}
            \Ffrak[f] = \fhat(w) = \int_{\Rbb} f(x) e^{-i w x} \diff x
        \end{align*}
    $$
</div>
<p>&#x4E8E;&#x662F;</p>
<div>
    $$
        \begin{align*}
            \lim_{N \rightarrow \infty} \int_{-N}^N \fhat(w) e^{i w y} \diff w &amp; = 2 \pi \int_{\Rbb} f(y+z) \left[ \lim_{N \rightarrow \infty} \frac{\sin (N z)}{\pi z} \right] \diff z \\
            &amp; = 2 \pi \int_{\Rbb} f(y+z) \delta(z) \diff z = 2 \pi f(y)
        \end{align*}
    $$
</div>
<p>&#x6545;&#x5BF9;&#x5E94;&#x7684; Fourier &#x9006;&#x53D8;&#x6362;&#x4E3A;</p>
<div>
    $$
        \begin{align*}
            \Ffrak^{-1}[\fhat] = f(y) = \frac{1}{2\pi} \int_\Rbb \fhat(w) e^{i w y} \diff w
        \end{align*}
    $$
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        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
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</section></section><section><section data-notes lineno="213" class="slide " data-line="213" data-h="2" data-v="0">
<div class="multi_column">
    <div class="title_hr"> 
        <hr class="hr_top">
        <h5 class="title">&#x5377;&#x79EF;&#x5B9A;&#x7406;</h5>
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<p>&#x51FD;&#x6570;<span class="mathjax-exps">$f$</span>&#x3001;<span class="mathjax-exps">$g$</span>&#x7684;&#x5377;&#x79EF;</p>
<div>
    $$
        \begin{align*}
            (f*g) (x) = \int_{\Rbb} f(y) g(x - y) \diff y
        \end{align*}
    $$
</div>
<p>&#x7684; Fourier &#x53D8;&#x6362;&#x7B49;&#x4E8E;&#x5176;&#x5404;&#x81EA; Fourier &#x53D8;&#x6362;&#x7684;&#x4E58;&#x79EF;</p>
<div>
    $$
        \begin{align*}
            \Ffrak &amp; [f * g] (w) = \int_\Rbb \left[ (f*g)(x) \right] \exp(-i w x) \diff x                                                                           \\
            &amp; = \int_\Rbb \left[ \int_\Rbb f(y) g(x-y) \diff y \right] \exp(-i w x) \diff x                                                                           \\
            &amp; = \int_\Rbb \underbrace{\left[ \int_\Rbb g(x - y) \exp (- i w (x - y) \diff x \right]}_{\Ffrak[g](w)} f(y) \exp (-i w y) \diff y \\
            &amp; = \Ffrak[f](w) \cdot \Ffrak[g](w) \qquad \Longrightarrow f * g = \Ffrak^{-1}(\Ffrak[f]\cdot \Ffrak[g])
        \end{align*}
    $$
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</section><section vertical="true" data-notes lineno="254" class="slide " data-line="254" data-h="2" data-v="1">
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<p>&#x5377;&#x79EF;&#x7684;&#x5E73;&#x79FB;&#x4E0D;&#x53D8;&#x6027;&#xFF1A;</p>
<div>
    $$
        \begin{align*}
            (f*g) (t + a) &amp; = \int_{\Rbb} f(y) g(t + a - y) \diff y = f(t) * g(t+a) \\
            &amp; = \int_{\Rbb} f(x+a) g(t - x) \diff x = f(t+a) * g(t)
        \end{align*}
    $$
</div>
<br>
<p>&#x5377;&#x79EF;&#x7684;&#x5DEE;&#x5206;&#x4E0D;&#x53D8;&#x6027;&#xFF1A;</p>
<div>
    $$
        \begin{align*}
            \Delta(f*g) (t) &amp; = (f*g) (t+1) - (f*g) (t) = \\
            &amp; = \int_{\Rbb} f(y) g(t + 1 - y) \diff y - \int_{\Rbb} f(y) g(t - y) \diff y \\
            &amp; = \int_{\Rbb} f(y) \Delta g(t - y) \diff y = f(t) * \Delta g(t)
        \end{align*}
    $$
</div>
<br>
<p>&#x540C;&#x7406;&#x53EF;&#x8BC1;<span class="mathjax-exps">$\Delta(f*g) (t) = \Delta f(t) * g(t)$</span></p>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x8FDB;&#x9636;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="301" class="slide " data-line="301" data-h="2" data-v="2">
<div class="multi_column">
    <div class="title_hr"> 
        <hr class="hr_top">
        <h5 class="title">&#x79BB;&#x6563;&#x5377;&#x79EF;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x82E5;&#x53EA;&#x6709;<span class="mathjax-exps">$f(x)$</span>&#x5728;&#x6709;&#x9650;&#x70B9;&#x4E0A;&#x7684;&#x503C;&#xFF1A;<span class="mathjax-exps">$f(1), f(2), \ldots, f(N)$</span>&#xFF0C;&#x53EF;&#x4EE5;&#x8003;&#x8651;&#x79BB;&#x6563;&#x5377;&#x79EF;</p>
<div>
    $$
        \begin{align*}
            (f *_N g)(t) = \sum_{\tau \in [N]} f(\tau) g( (t - \tau)_{\modd N})
        \end{align*}
    $$
</div>
<br>
<p>&#x4ECE;&#x4FE1;&#x53F7;&#x5904;&#x7406;&#x7684;&#x89D2;&#x5EA6;&#x770B;&#xFF0C;&#x79BB;&#x6563;&#x5377;&#x79EF;&#x662F;&#x5BF9;&#x65F6;&#x5E8F;&#x4FE1;&#x53F7;<span class="mathjax-exps">$f(1), f(2), \ldots, f(N)$</span>&#x7684;&#x6EE4;&#x6CE2;&#x64CD;&#x4F5C;&#xFF0C;&#x5377;&#x79EF;&#x51FD;&#x6570;<span class="mathjax-exps">$g$</span>&#x79F0;&#x4E3A;&#x6EE4;&#x6CE2;&#x5668;(filter)&#xFF0C;&#x4F8B;&#x5982;&#x9AD8;&#x901A;&#x3001;&#x4F4E;&#x901A;&#x3001;&#x5E26;&#x901A;</p>
<br>
<p>&#x6EE4;&#x6CE2;&#x548C;&#x5E73;&#x79FB;&#x4E0D;&#x53D8;&#x6027;&#x662F;&#x6570;&#x5B57;&#x4FE1;&#x53F7;&#x5904;&#x7406;&#x9886;&#x57DF;(digital signal processing, DSP)&#x7684;&#x6838;&#x5FC3;&#xFF0C;&#x4E5F;&#x542F;&#x53D1;&#x4E86;&#x540E;&#x6765;&#x7684;&#x5377;&#x79EF;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;</p>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x8FDB;&#x9636;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section></section><section><section data-notes lineno="337" class="slide " data-line="337" data-h="3" data-v="0">
<div class="multi_column">
    <div class="title_hr"> 
        <hr class="hr_top">
        <h5 class="title">&#x65F6;&#x5E8F;&#x4FE1;&#x53F7;$\longrightarrow$&#x56FE;&#x4FE1;&#x53F7;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x65F6;&#x5E8F;&#x4FE1;&#x53F7;<span class="mathjax-exps">$f(1), f(2), \ldots, f(N)$</span>&#x53EF;&#x4EE5;&#x770B;&#x4F5C;<span class="mathjax-exps">$N$</span>&#x4E2A;&#x70B9;&#x7684;&#x6709;&#x5411;&#x5706;&#x73AF;(&#x56FE;)&#xFF1A;</p>
<ul>
<li>&#x65F6;&#x95F4;<span class="mathjax-exps">$t$</span>&#x5BF9;&#x5E94;&#x7B2C;<span class="mathjax-exps">$t$</span>&#x4E2A;&#x70B9;&#xFF0C;<span class="mathjax-exps">$f(t)$</span>&#x5C31;&#x662F;<span class="mathjax-exps">$t$</span>&#x65F6;&#x523B;&#x7684;&#x4FE1;&#x53F7;&#xFF0C;&#x8BB0;&#x4F5C;<span class="mathjax-exps">$\fv = [f(1); f(2); \ldots; f(N)]$</span></li>
<li>&#x7B2C;<span class="mathjax-exps">$1$</span>&#x4E2A;&#x70B9;&#x6307;&#x5411;&#x7B2C;<span class="mathjax-exps">$2$</span>&#x4E2A;&#x70B9;&#xFF0C;&#x7B2C;<span class="mathjax-exps">$2$</span>&#x4E2A;&#x70B9;&#x6307;&#x5411;&#x7B2C;<span class="mathjax-exps">$3$</span>&#x4E2A;&#x70B9;&#xFF0C;&#x2026;&#x2026;&#xFF0C;&#x7B2C;<span class="mathjax-exps">$N$</span>&#x4E2A;&#x70B9;&#x6307;&#x5411;&#x7B2C;<span class="mathjax-exps">$1$</span>&#x4E2A;&#x70B9;</li>
</ul>
<br>
<p><span class="mathjax-exps">$N$</span>&#x4E2A;&#x70B9;&#x7684;&#x6709;&#x5411;&#x5706;&#x73AF;&#x7684;&#x90BB;&#x63A5;&#x77E9;&#x9635;&#x548C; Laplace &#x77E9;&#x9635;&#x5206;&#x522B;&#x4E3A;&#xFF1A;</p>
<div>
    $$
        \begin{align*}
            \Av_c = \begin{bmatrix} 0 &amp; 1 \\ &amp; 0 &amp; 1 \\ &amp; &amp; 0 \\ &amp; &amp; &amp; \ddots \\ &amp; &amp; &amp; &amp; 0 &amp; 1 \\ 1 &amp; &amp; &amp; &amp; &amp; 0 \end{bmatrix}, \quad \Lv_c = \begin{bmatrix} 1 &amp; -1 \\ &amp; 1 &amp; -1 \\ &amp; &amp; 1 \\ &amp; &amp; &amp; \ddots \\ &amp; &amp; &amp; &amp; 1 &amp; -1 \\ -1 &amp; &amp; &amp; &amp; &amp; 1 \end{bmatrix}
        \end{align*}
    $$
</div>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x8FDB;&#x9636;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="372" class="slide " data-line="372" data-h="3" data-v="1">
<div class="multi_column">
    <div class="title_hr"> 
        <hr class="hr_top">
        <h5 class="title">&#x65F6;&#x5E8F;&#x4FE1;&#x53F7;$\longrightarrow$&#x56FE;&#x4FE1;&#x53F7;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x5BF9;&#x4E8E;&#x4FE1;&#x53F7;&#xFF1A;<span class="mathjax-exps">$\fv = [f_1; f_2; \ldots; f_N]$</span>&#xFF0C;&#x6709;</p>
<div>
    $$
        \begin{align*}
            \Av_c \fv = \begin{bmatrix} f_2 \\ f_3 \\ \vdots \\ f_1 \end{bmatrix}, \quad \Lv_c \fv = \begin{bmatrix} f_1 - f_2 \\ f_2 - f_3 \\ \vdots \\ f_N - f_1 \end{bmatrix}
        \end{align*}
    $$
</div>
<p>&#x5373;<span class="mathjax-exps">$[\Av_c \fv]_t = [\fv]_{t_{\modd N} + 1}$</span>&#xFF0C;<span class="mathjax-exps">$[\Lv_c \fv]_t = [\fv]_t - [\fv]_{t_{\modd N} + 1}$</span></p>
<br>
<p>&#x5C06;&#x4FE1;&#x53F7;&#x770B;&#x4F5C;&#x6709;&#x5411;&#x5706;&#x73AF;&#x540E;&#xFF0C;&#x4FE1;&#x53F7;&#x7684;&#x5E73;&#x79FB;&#x64CD;&#x4F5C;&#x548C;&#x5DEE;&#x5206;&#x64CD;&#x4F5C;&#x53EF;&#x4EE5;&#x5206;&#x522B;&#x901A;&#x8FC7;&#x5DE6;&#x4E58;&#x90BB;&#x63A5;&#x77E9;&#x9635;&#x548C;Laplace&#x77E9;&#x9635;&#x5F97;&#x5230;</p>
<br>
<p>&#x65F6;&#x5E8F;&#x4FE1;&#x53F7; <span class="mathjax-exps">$\longleftrightarrow$</span> &#x5377;&#x79EF;&#x51FD;&#x6570;<br>
&#x56FE;&#x4E0A;&#x4FE1;&#x53F7; <span class="mathjax-exps">$\longleftrightarrow$</span> &#x56FE;&#x7684;&#x77E9;&#x9635;</p>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x8FDB;&#x9636;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="411" class="slide " data-line="411" data-h="3" data-v="2">
<div class="multi_column">
    <div class="title_hr"> 
        <hr class="hr_top">
        <h5 class="title">&#x4E00;&#x822C;&#x60C5;&#x5F62;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x5BF9;&#x4E8E;&#x4E00;&#x822C;&#x7684;&#x5377;&#x79EF;&#x64CD;&#x4F5C;&#xFF0C;&#x53EF;&#x4EE5;&#x671F;&#x671B;<span class="mathjax-exps">$f * g = \Qv_g \fv$</span>&#xFF0C;&#x5176;&#x4E2D;&#x77E9;&#x9635;<span class="mathjax-exps">$\Qv_g \in \Rbb^{N \times N}$</span>&#x662F;&#x5377;&#x79EF;&#x51FD;&#x6570;<span class="mathjax-exps">$g$</span>&#x5BF9;&#x5E94;&#x7684;&#x77E9;&#x9635;&#x8868;&#x793A;</p>
<br>
<p><span class="mathjax-exps">$\Qv_g$</span>&#x9700;&#x6EE1;&#x8DB3;&#x7684;&#x6761;&#x4EF6;</p>
<ul>
<li>&#x5377;&#x79EF;&#x7684;&#x5E73;&#x79FB;&#x4E0D;&#x53D8;&#x6027;&#xFF1A;<span class="mathjax-exps">$\Av_c \Qv_g = \Qv_g \Av_c$</span></li>
<li>&#x5377;&#x79EF;&#x7684;&#x5DEE;&#x5206;&#x4E0D;&#x53D8;&#x6027;&#xFF1A;<span class="mathjax-exps">$\Lv_c \Qv_g = \Qv_g \Lv_c$</span></li>
</ul>
<br>
<p>&#x7531;&#x77E9;&#x9635;&#x4E58;&#x6CD5;&#x7684;&#x7ED3;&#x5408;&#x5F8B;&#x53EF;&#x53D6;&#x4E0B;&#x8FF0;&#x591A;&#x9879;&#x5F0F;&#x5F62;&#x5F0F;</p>
<div>
    $$
        \begin{align*}
            \Qv_g = p_N(\Av) = \alpha_0 \Iv + \alpha_1 \Av_c + \cdots + \alpha_{N-1} \Av_c^{N-1}
        \end{align*}
    $$
</div>
<br>
<p>&#x8FDB;&#x4E00;&#x6B65;&#xFF0C;&#x4E00;&#x822C;&#x56FE;&#x4E0A;&#x7684;&#x5377;&#x79EF;&#x6EE4;&#x6CE2;&#x5668;&#x5BF9;&#x5E94;&#x7684;&#x77E9;&#x9635;&#x8868;&#x793A;&#x4E3A;</p>
<div>
    $$
        \begin{align*}
            \Qv_g = \alpha_0 \Iv + \alpha_1 \Av + \cdots + \alpha_{N-1} \Av^{N-1}
        \end{align*}
    $$
</div>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x8FDB;&#x9636;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="462" class="slide " data-line="462" data-h="3" data-v="3">
<div class="multi_column">
    <div class="title_hr"> 
        <hr class="hr_top">
        <h5 class="title">&#x4E00;&#x822C;&#x60C5;&#x5F62;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x4E00;&#x822C;&#x56FE;&#x4E0A;&#x7684;&#x5377;&#x79EF;&#x6EE4;&#x6CE2;&#x5668;&#x5BF9;&#x5E94;&#x7684;&#x77E9;&#x9635;&#x8868;&#x793A;&#x4E3A;</p>
<div>
    $$
        \begin{align*}
            \Qv_g = \alpha_0 \Iv + \alpha_1 \Av + \cdots + \alpha_{N-1} \Av^{N-1}
        \end{align*}
    $$
</div>
<br>
<p>&#x8BBE;&#x6BCF;&#x4E2A;&#x70B9;&#x6709;<span class="mathjax-exps">$1$</span>&#x7EF4;&#x7684;&#x7279;&#x5F81;&#xFF0C;&#x6784;&#x6210;&#x5411;&#x91CF;<span class="mathjax-exps">$\xv \in \Rbb^{|\Vcal|}$</span>&#xFF0C;&#x4E8E;&#x662F;</p>
<div>
    $$
        \begin{align*}
            [\Qv_g \xv]_u = \alpha_0 [\Iv \xv]_u + \alpha_1 [\Av \xv]_u + \cdots + \alpha_{N-1} [\Av^{N-1} \xv]_u
        \end{align*}
    $$
</div>
<p>&#x5373;&#x5377;&#x79EF;&#x4E00;&#x6B21;&#x540E;&#xFF0C;&#x70B9;<span class="mathjax-exps">$u$</span>&#x7684;&#x4FE1;&#x606F;&#x53D8;&#x4E3A;&#x81EA;&#x8EAB;&#x3001;<span class="mathjax-exps">$1$</span>&#x9636;&#x90BB;&#x5C45;&#x3001;&#x2026;&#x2026;&#x3001;<span class="mathjax-exps">$N-1$</span>&#x9636;&#x90BB;&#x5C45;&#x7684;&#x4FE1;&#x606F;&#x7684;&#x7EBF;&#x6027;&#x7EC4;&#x5408;</p>
<br>
<p>&#x82E5;&#x6BCF;&#x4E2A;&#x70B9;&#x6709;<span class="mathjax-exps">$d$</span>&#x7EF4;&#x7279;&#x5F81;&#xFF0C;&#x6784;&#x6210;&#x77E9;&#x9635;<span class="mathjax-exps">$\Xv \in \Rbb^{|\Vcal| \times d}$</span>&#xFF0C;&#x4E8E;&#x662F;</p>
<div>
    $$
        \begin{align*}
            \Qv_g \Xv = \alpha_0 \Iv \Xv + \alpha_1 \Av \Xv + \cdots + \alpha_{N-1} \Av^{N-1} \Xv
        \end{align*}
    $$
</div>
<br>
<p>&#x4ECE;&#x4FE1;&#x53F7;&#x5904;&#x7406;&#x7684;&#x89D2;&#x5EA6;&#x770B;&#xFF0C;&#x4E0D;&#x540C;&#x7EF4;&#x5EA6;&#x7684;&#x7279;&#x5F81;&#x4E5F;&#x79F0;&#x4E3A;&#x201C;&#x901A;&#x9053;&#x201D;(channel)</p>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x8FDB;&#x9636;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section><section vertical="true" data-notes lineno="520" class="slide " data-line="520" data-h="3" data-v="4">
<div class="multi_column">
    <div class="title_hr"> 
        <hr class="hr_top">
        <h5 class="title">&#x4E00;&#x822C;&#x60C5;&#x5F62;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x4E00;&#x822C;&#x56FE;&#x4E0A;&#x7684;&#x5377;&#x79EF;&#x6EE4;&#x6CE2;&#x5668;&#x5BF9;&#x5E94;&#x7684;&#x77E9;&#x9635;&#x8868;&#x793A;&#x4E3A;</p>
<div>
    $$
        \begin{align*}
            \Qv_g = \alpha_0 \Iv + \alpha_1 \Av + \cdots + \alpha_{N-1} \Av^{N-1}
        \end{align*}
    $$
</div>
<br>
<p>&#x5BF9;&#x4E8E;&#x6D88;&#x606F;&#x4F20;&#x9012;&#x6846;&#x67B6;</p>
<ul>
<li>&#x6BCF;&#x5C42;&#x91C7;&#x7528;&#x7684;&#x662F;&#x7B80;&#x5355;&#x5377;&#x79EF;&#x6838;<span class="mathjax-exps">$\Qv_g = \Iv + \Av$</span>&#xFF0C;&#x5373;&#x53EA;&#x8003;&#x8651;<span class="mathjax-exps">$1$</span>&#x9636;&#x90BB;&#x5C45;&#x7684;&#x4FE1;&#x606F;</li>
<li>&#x901A;&#x8FC7;&#x5806;&#x53E0;&#x591A;&#x5C42;&#xFF0C;&#x9690;&#x5F0F;&#x5730;&#x5F15;&#x5165;&#x4E86;&#x9AD8;&#x9636;&#x7684;&#x90BB;&#x63A5;&#x77E9;&#x9635;</li>
</ul>
<br>
<p>&#x5BF9;&#x4E8E;&#x6709;&#x5411;&#x5706;&#x73AF;&#xFF0C;&#x5EA6;&#x77E9;&#x9635;<span class="mathjax-exps">$\Dv = \Iv$</span>&#xFF0C;&#x56E0;&#x6B64; Laplace &#x77E9;&#x9635;<span class="mathjax-exps">$\Lv = \Iv - \Av$</span>&#xFF0C;&#x4ECE;&#x800C;<span class="mathjax-exps">$p_N(\Av)$</span>&#x53EF;&#x4EE5;&#x4FDD;&#x8BC1;&#x5DEE;&#x5206;&#x4E0D;&#x53D8;&#x6027;&#xFF0C;&#x5373;<span class="mathjax-exps">$\Lv \cdot p_N(\Av) = p_N(\Av) \cdot \Lv$</span>&#xFF0C;&#x4F46;&#x8BE5;&#x7ED3;&#x8BBA;&#x5BF9;&#x4E00;&#x822C;&#x56FE;&#x4E0D;&#x6210;&#x7ACB;</p>
<br>
<p>&#x5BF9;&#x4E8E;&#x4E00;&#x822C;&#x56FE;&#xFF0C;&#x901A;&#x5E38;&#x8003;&#x8651;<span class="mathjax-exps">$\Lv_\sym = \Dv^{-1/2} \Lv \Dv^{-1/2}$</span>&#x3001;<span class="mathjax-exps">$\Av_\sym = \Dv^{-1/2} \Av \Dv^{-1/2}$</span>&#x7684;&#x591A;&#x9879;&#x5F0F;&#xFF0C;&#x597D;&#x5904;&#x662F;<span class="mathjax-exps">$\Lv_\sym = \Dv^{-1/2} (\Dv - \Av) \Dv^{-1/2} = \Iv - \Av_\sym$</span>&#xFF0C;&#x4ECE;&#x800C;&#x4E24;&#x4E2A;&#x4E0D;&#x53D8;&#x5F62;&#x4F9D;&#x7136;&#x53EF;&#x4EE5;&#x540C;&#x65F6;&#x6EE1;&#x8DB3;</p>
<div class="footer">
    <hr class="hr_bottom">
    <div class="multi_column">
        <h6 class="bottom_left">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x5BFC;&#x8BBA;</h6>
        <h6 class="bottom_center">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x8FDB;&#x9636;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
    </div>
</div>
</section></section><section><section data-notes lineno="563" class="slide " data-line="563" data-h="4" data-v="0">
<div class="multi_column">
    <div class="title_hr"> 
        <hr class="hr_top">
        <h5 class="title">&#x8C31;&#x56FE;&#x5377;&#x79EF;</h5>
    </div>
    <img class="xiaohui" src="../common/img/xiaohui.png" height="120px">
</div>
<p>&#x81F3;&#x6B64;&#x6211;&#x4EEC;&#x5DF2;&#x5C06;&#x4FE1;&#x53F7;&#x548C;&#x5377;&#x79EF;&#x7684;&#x6982;&#x5FF5;&#x63A8;&#x5E7F;&#x5230;&#x56FE;&#x4E0A;&#xFF0C;&#x8FD8;&#x5DEE;&#x4E00;&#x4E2A; Fourier &#x53D8;&#x6362;</p>
<p>Laplace &#x7B97;&#x5B50;</p>
<div>
    $$
        \begin{align*}
            \Delta f = \div (\grad f) = \nabla \cdot (\nabla f) = \sum_{i \in [d]} \frac{\partial^2 f}{\partial x_i^2}
        \end{align*}
    $$
</div>
<p>&#x4E8E;&#x662F;<span class="mathjax-exps">$\Delta (e^{-iwx}) = (i w)^2 e^{-iwx} = -w^2 e^{-iwx}$</span>&#xFF0C;&#x5373; Laplace &#x7B97;&#x5B50;&#x7684;&#x7279;&#x5F81;&#x51FD;&#x6570;&#x662F;<span class="mathjax-exps">$e^{-iwx}$</span>&#xFF0C;&#x5BF9;&#x5E94;&#x7279;&#x5F81;&#x503C;&#x662F;<span class="mathjax-exps">$-w^2$</span>&#xFF0C;&#x800C;<span class="mathjax-exps">$e^{-iwx}$</span>&#x6070;&#x597D;&#x662F; Fourier &#x53D8;&#x6362;&#x7528;&#x5230;&#x7684;&#x57FA;&#x51FD;&#x6570;</p>
<br>
<p>&#x7C7B;&#x6BD4;&#x8FC7;&#x6765;&#xFF0C;Laplace &#x77E9;&#x9635;&#x7684;&#x7279;&#x5F81;&#x5411;&#x91CF;&#x5C31;&#x662F;&#x56FE;&#x4E0A;&#x7684; Fourier &#x53D8;&#x6362;&#x7684;&#x57FA;&#x5411;&#x91CF;&#xFF0C;&#x8BBE;<span class="mathjax-exps">$\Lv$</span>&#x7684;&#x7279;&#x5F81;&#x503C;&#x5206;&#x89E3;&#x4E3A;<span class="mathjax-exps">$\Lv = \Uv \Lambdav \Uv^\top$</span>&#xFF0C;&#x4E8E;&#x662F;&#x56FE;&#x4E0A;&#x7684; Fourier &#x53D8;&#x6362;&#x548C;&#x9006;&#x53D8;&#x6362;&#x5206;&#x522B;&#x4E3A;</p>
<div>
    $$
        \begin{align*}
            \sv = \Uv^\top \fv, \quad \fv = \Uv \sv
        \end{align*}
    $$
</div>
<br>
<p>&#x5377;&#x79EF;&#x5B9A;&#x7406;&#xFF1A;<span class="mathjax-exps">$\fv *_\Gcal \gv = \Uv (\Uv^\top \fv \odot \Uv^\top \gv)$</span></p>
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        <h6 class="bottom_center">&#x56FE;&#x795E;&#x7ECF;&#x7F51;&#x7EDC;&#x8FDB;&#x9636;</h6>
        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
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</section><section vertical="true" data-notes lineno="611" class="slide " data-line="611" data-h="4" data-v="1">
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    <div class="title_hr"> 
        <hr class="hr_top">
        <h5 class="title">&#x8C31;&#x56FE;&#x5377;&#x79EF;</h5>
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</div>
<p>&#x5377;&#x79EF;&#x5B9A;&#x7406;&#xFF1A;<span class="mathjax-exps">$\fv *_\Gcal \gv = \Uv (\Uv^\top \fv \odot \Uv^\top \gv)$</span></p>
<br>
<p>&#x5F15;&#x5165; Fourier &#x7CFB;&#x6570;<span class="mathjax-exps">$\thetav_g = \Uv^\top \gv$</span>&#xFF0C;&#x6211;&#x4EEC;&#x76F4;&#x63A5;&#x6709;</p>
<div>
    $$
        \begin{align*}
            \fv *_\Gcal \gv = \Uv (\Uv^\top \fv \odot \thetav_g) = (\Uv \diag(\thetav_g) \Uv^\top) \fv
        \end{align*}
    $$
</div>
<br>
<p>&#x5982;&#x679C;&#x5BF9;&#x5BF9;&#x89D2;&#x9635;<span class="mathjax-exps">$\diag(\thetav_g)$</span>&#x7684;&#x9009;&#x62E9;&#x4E0D;&#x52A0;&#x9650;&#x5236;&#x53EF;&#x80FD;&#x4F1A;&#x4E0D;&#x6EE1;&#x8DB3;&#x5377;&#x79EF;&#x5E94;&#x6709;&#x7684;&#x826F;&#x597D;&#x6027;&#x8D28;&#xFF0C;&#x901A;&#x5E38;&#x53EF;&#x53D6;<span class="mathjax-exps">$\diag(\thetav_g) = p_N(\Lambdav)$</span>&#xFF0C;&#x4E8E;&#x662F;</p>
<div>
    $$
        \begin{align*}
            \fv *_\Gcal \gv = (\Uv p_N(\Lambdav) \Uv^\top) \fv = p_N(\Lv) \fv
        \end{align*}
    $$
</div>
<br>
<p>&#x8FD9;&#x79F0;&#x4E3A;&#x9891;&#x57DF;&#x89C6;&#x89D2;&#x7684; GCN&#xFF0C;&#x800C;&#x524D;&#x9762;&#x4E0D;&#x91C7;&#x7528; Fourier &#x53D8;&#x6362;&#x5BFC;&#x51FA;&#x7684;&#x662F;&#x7A7A;&#x57DF;&#x89C6;&#x89D2;&#x7684; GCN</p>
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        <h6 class="bottom_right">tengzhang@hust.edu.cn</h6>
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